This page is intended to be the best-available compendium of information about the probabilities
of different pathologies in 3-candidate instant runoff voting
(IRV) elections.
The "total paradox probability"
in such elections, i.e. the probability that at least one among
the 8 pathologies {Q, R, U, V, W, X, Y, Z} listed
below (note that set does not include
S and T) occur in a random election,
is found to be

24.59%, 13.99%, and 27.50%

in our three different probability models (defined
below) respectively.
But if we restrict attention to elections in which the IRV process matters,
i.e. in which the IRV and plain-plurality
winners differ (i.e. exactly the elections IRV-advocates
tend to cite as examples of the "success" of the Instant Runoff Voting process),
the total paradox probability
becomes stunningly large:

74.10%, 72.61%, and 54.44%

For the most part, this was not previously recognized.
This goes a long way toward explaining why it has been so incredibly easy for people like
me to find pathologies in real-world IRV elections, seemingly most of the time we ever looked
at any interesting IRV election for which we could obtain enough data, and
seemingly especially
in the elections cited by IRV-advocates as "great successes" for IRV.

These numbers appear a rather serious indictment of Instant Runoff Voting
as a decision-making process, as well as hurting its enactment/repeal chances.
These large probabilities differ from the perception advanced by the IRV-advocacy
group "FairVote," formerly named the "Center for Voting and Democracy."
E.g. FairVote's "senior analyst" Steven Hill and the head of Fairvote,
Rob Richie, are quoted in the 2008 popular science book
Gaming the Vote
by William Poundstone (page 268), thus:

[FairVote's] position is that such paradoxes are too rare to worry about.
"We've had thousands of [IRV] elections and it's not an issue,"
said Rob Richie.
Steven Hill, a senior analyst with [FairVote] dismisses "these mathematical paradoxes that
while in theory are interesting for mathematicians to doodle around on their sketch pads,
in fact have no basis in reality... it's also possible that a meteorite will strike
Earth and wipe out life... but probably not for a few more million years."

FairVote, Richie, and Hill have all cited various specific IRV elections as "great
successes," including
Burlington Vermont USA 2009,
Ireland 1990, and the
Australian House elections of 2007,
which actually contained pathologies such as nonmonotonicity
(a fact they, during these laudings, left unmentioned).
Indeed, despite me and others pointing out these and other examples
to FairVote repeatedly for many years,
they continue to maintain
to this very day

In terms of the frequency of non-monotonicity in real-world elections:
there is no evidence that this has ever played a role in any IRV election –
not the IRV presidential elections in Ireland, nor the literally thousands of hotly contested IRV federal elections that have taken place for generations in Australia, nor in any of the IRV elections in the United States.
–
FairVote
webpage titled
"Monotonicity and IRV – Why the Monotonicity Criterion is of Little Import"
downloaded 24 August 2010; our emphasis added.

1. The three probability models

View a V-voter C-candidate election as a big table with V rows.
The kth row specifies which of the C! possible rank-order-style ballots
was the vote selected by the kth voter. We assume all
among the C!V possible such tables are equally likely, and take the limit
V→∞.

Dirichlet model
(also has been called "impartial anonymous culture"):

The election is viewed as being defined by its vector of the C! nonnegative integer
"vote totals" (one total for each possible ordering of the candidates, i.e.
saying how many voters provided that ordering); this vector sums to V, the total number of voters.
All such vectors are considered equally likely, and we
take the limit
V→∞.

Quas's "1D political spectrum" model:

The C candidates are regarded as independent random-uniform points on the real interval (0,1).
The voters are the uniform distribution on (0,1) itself (the candidates are samples from it).
As their vote, each voter orders the candidates in order of increasing distance.
For example,

0------------J-----*------------K----------L----------1

in this picture, the candidates dropped randomly on the line segment (0,1)
are named J, K, and L; and the voter located at * votes "J>K>L."

2. Mathematical techniques

In the Quas and Dirichlet models the probability of any particular event defined
by linear inequalities is proportional to a
polytope volume in a C-dimensional or (C!-1)-dimensional space, respectively.
It is possible to compute all these as exact rational numbers, in principle.
For finite instead of infinite V,
it comes down to counting lattice points inside polytopes, and that is in principle
doable exactly as an "Ehrhart polynomial(V)" formula.
In the REM, the probabilities instead arise from polytope volumes in a
nonEuclidean spherical geometry. These are also in principle expressible exactly
in terms of "Schläfli functions."

However, I'm not doing any of that here.
I'm just using Monte Carlo inexact evaluation.

However, it looks to me now that all the Quas 1D probabilities tabulated in the blue table
can be written as rational numbers with denominator=360,
and all those in the yellow table with denominator=1800.
I.e. our inexact Quas evaluations,
if these conjectures are true (and they appear to be) can trivially be converted
into exact ones.
For example, the approximate number "6.9446" from the blue Quas table is 2500.056/360
which presumably means the corresponding exact answer is 2500/360=125/18=6.94444...
and in fact this is true.

The Monte Carlo data is presently insufficiently precise to determine the corresponding magic
denominators for the Dirichlet model, but it strongly supports the idea that
for the blue table, it is some multiple of 41472,
while for the yellow table, it is a multiple of 12780.

3. Master List of Configurations and their Probabilities

Consider the following 10
possible paradoxes that could afflict a 3-candidate Instant Runoff Voting
(IRV) election with
winner A and second-place finisher B.

In cases S and T, a better word than "paradox" or "pathology"
might be "phenomenon."
That is, S and T cannot be regarded as "flaws" in the IRV election process, they
are just phenomena that would have happened with those votes for any election process.
That is why our "total paradox probability" includes only
the 8 pathologies {Q, R, U, V, W, X, Y, Z} and not either S or T.
Professor Jack Nagel has argued R also should have been un-included. For that, see
§5.

"Loser drop-out paradox." An IRV-loser, by dropping out of the race, would change the winner.
(The loser, also called a "spoiler," is B, and when B drops out the winner changes from A to C.)
The same set of 3-candidate IRV
elections also exhibit "Favorite betrayal paradox": a set of co-voting
voters are better off "betraying their favorite" than honestly voting that favorite top,
i.e. it is unstrategic to vote for a "spoiler."

All scoring rules
(including Plain Plurality, antiPlurality, Borda, etc) agree the winner is B,
disagreeing with IRV's winner A. If you agree that "democracy" should elect
the most-loved and least-hated candidate, then since in these situations
B is both of those things simultaneously,
you must agree IRV got the wrong winner in these situations.

There are 1024
possible combinations of those phenomena or their negations, which we denote
by a 10-bit binary string.
For example 0000100010 means "U and Y occur
but QRST_VWX_Z all do not."
We list every possible 10-bit string along with an IRV election example allegedly
yielding precisely that combination of pathologies –
and the probability of occurrence of that combination in our three probability models.
(Bitstrings omitted if that configuration impossible, which is why there are many fewer than
1024 lines in our table.)
Each percentage in the next two tables is based on
25×1010
Monte-Carlo experiments and therefore has additive
standard error
±0.0001
or closer, i.e. please read "33.3333%" as "(33.3333±0.0001)%."
The total computer time (2006-era personal computer) was
≈30 hours.

QRSTUVWXYZ

Election Example

REM Prob.

Dirichlet Prob.

Quas-1D Prob.

0000000000

ABC= 0, ACB= 2, BAC= 0, BCA= 1, CAB= 0, CBA= 0

69.0762%

82.6390%

61.1110%

0001000000

ABC= 0, ACB= 7, BAC=12, BCA= 0, CAB= 6, CBA= 0

6.3361%

3.3757%

11.3890%

0001000100

ABC=15, ACB= 0, BAC=20, BCA= 0, CAB=10, CBA= 0

7.8076%

2.7826%

8.0555%

0001010100

ABC=20, ACB= 0, BAC=16, BCA=14, CAB=15, CBA= 0

2.8818%

0.9163%

0.0000%

0101000000

ABC=23, ACB=29, BAC=17, BCA=40, CAB= 6, CBA= 0

0.0164%

0.0096%

0.0000%

0101000100

ABC=18, ACB= 0, BAC= 8, BCA=15, CAB=10, CBA= 0

1.5332%

1.0079%

0.0000%

0101010100

ABC=26, ACB= 0, BAC=16, BCA=23, CAB=18, CBA= 0

0.1903%

0.0675%

0.0000%

1000000010

ABC= 0, ACB=13, BAC= 0, BCA=12, CAB= 6, CBA= 0

0.0000%

0.6970%

7.0001%

1000001010

ABC= 0, ACB=19, BAC= 0, BCA=18, CAB=12, CBA= 0

0.0000%

0.1087%

2.3750%

1000100010

ABC= 0, ACB=25, BAC= 6, BCA=18, CAB=18, CBA= 0

0.7542%

0.2211%

0.0000%

1000101010

ABC= 0, ACB=22, BAC= 0, BCA=21, CAB=16, CBA= 0

1.4849%

0.8830%

4.5139%

1001000010

ABC= 0, ACB=13, BAC= 4, BCA=14, CAB=10, CBA= 0

0.0000%

0.2796%

1.3333%

1001001010

ABC= 0, ACB=15, BAC= 0, BCA=20, CAB=10, CBA= 0

0.0000%

0.0866%

1.7916%

1001100010

ABC= 0, ACB=13, BAC= 6, BCA=12, CAB=12, CBA= 0

0.5816%

0.1912%

0.0000%

1001101010

ABC= 0, ACB=25, BAC= 0, BCA=30, CAB=20, CBA= 0

0.5637%

0.4842%

2.4305%

1010000000

ABC= 6, ACB= 7, BAC= 0, BCA=12, CAB= 6, CBA= 0

0.0000%

0.7006%

0.0000%

1010000001

ABC=13, ACB= 0, BAC= 0, BCA=12, CAB= 6, CBA= 0

0.0000%

0.8733%

0.0000%

1010001000

ABC= 9, ACB=10, BAC= 0, BCA=18, CAB=12, CBA= 0

0.0000%

0.0901%

0.0000%

1010001001

ABC=19, ACB= 0, BAC= 0, BCA=18, CAB=12, CBA= 0

0.0000%

0.0938%

0.0000%

1010100000

ABC=12, ACB=13, BAC= 6, BCA=18, CAB=18, CBA= 0

1.3645%

0.1487%

0.0000%

1010100001

ABC=25, ACB= 0, BAC= 6, BCA=18, CAB=11, CBA= 7

0.6750%

0.1497%

0.0000%

1010101000

ABC=11, ACB=11, BAC= 0, BCA=21, CAB=16, CBA= 0

1.5123%

0.5507%

0.0000%

1010101001

ABC=22, ACB= 0, BAC= 0, BCA=21, CAB=16, CBA= 0

0.6668%

0.5181%

0.0000%

1011000000

ABC= 5, ACB=11, BAC= 3, BCA=18, CAB=10, CBA= 0

0.0000%

0.4084%

0.0000%

1011000100

ABC= 8, ACB= 8, BAC= 3, BCA=18, CAB=10, CBA= 0

0.0000%

0.3064%

0.0000%

1011001000

ABC= 5, ACB=12, BAC= 0, BCA=23, CAB=11, CBA= 0

0.0000%

0.0677%

0.0000%

1011001100

ABC= 9, ACB= 8, BAC= 0, BCA=22, CAB=11, CBA= 0

0.0000%

0.0990%

0.0000%

1011010100

ABC=10, ACB=10, BAC= 6, BCA=24, CAB=15, CBA= 0

0.0000%

0.0672%

0.0000%

1011011100

ABC=10, ACB=10, BAC= 0, BCA=30, CAB=15, CBA= 0

0.0000%

0.0618%

0.0000%

1011100000

ABC=11, ACB=17, BAC= 9, BCA=24, CAB=22, CBA= 0

0.5674%

0.0468%

0.0000%

1011100100

ABC=14, ACB=14, BAC= 9, BCA=24, CAB=22, CBA= 0

0.9166%

0.0898%

0.0000%

1011101000

ABC=11, ACB=17, BAC= 0, BCA=33, CAB=22, CBA= 0

0.1713%

0.0848%

0.0000%

1011101100

ABC=14, ACB=14, BAC= 0, BCA=33, CAB=22, CBA= 0

0.3934%

0.2041%

0.0000%

1011110100

ABC=15, ACB=15, BAC=11, BCA=29, CAB=25, CBA= 0

0.9354%

0.1557%

0.0000%

1011111100

ABC=15, ACB=15, BAC= 0, BCA=40, CAB=25, CBA= 0

0.4667%

0.3326%

0.0000%

1111000101

ABC=16, ACB= 0, BAC= 3, BCA=18, CAB=10, CBA= 0

0.0000%

0.5566%

0.0000%

1111001101

ABC=17, ACB= 0, BAC= 0, BCA=22, CAB=11, CBA= 0

0.0000%

0.0738%

0.0000%

1111010101

ABC=20, ACB= 0, BAC= 6, BCA=24, CAB=15, CBA= 0

0.0000%

0.0823%

0.0000%

1111011101

ABC=20, ACB= 0, BAC= 0, BCA=30, CAB=15, CBA= 0

0.0000%

0.0347%

0.0000%

1111100101

ABC=28, ACB= 0, BAC= 9, BCA=24, CAB=16, CBA= 6

0.4807%

0.0665%

0.0000%

1111101101

ABC=28, ACB= 0, BAC= 0, BCA=33, CAB=22, CBA= 0

0.1528%

0.1374%

0.0000%

1111110101

ABC=30, ACB= 0, BAC=11, BCA=29, CAB=20, CBA= 5

0.3003%

0.0807%

0.0000%

1111111101

ABC=30, ACB= 0, BAC= 0, BCA=40, CAB=25, CBA= 0

0.1708%

0.1689%

0.0000%

Below is a smaller summary table of some of the most-requested pathology-probability
information in our three probability-models
(All of the numbers in the below tables are derivable by adding up
appropriate sets of numbers from the above master table):

X: Would be strategic mistake for more voters of some single type to come

16.2296%

7.2916%

8.0555%

T: Plurality and IRV winners differ

24.4661%

12.3263%

25.0000%

S: Condorcet cycle

8.7740%

6.2500%

0.0000%

Q∪R∪U∪V∪W∪X∪Y∪Z ("total paradox probability")

24.5877%

13.9853%

27.5000%

Both kinds of participation failure simultaneously W∩X

1.1837%

1.1121%

0.0000%

Both kinds of nonmonotonicity simultaneously U∩V

1.8732%

0.7378%

0.0000%

Q∪V: Betraying B makes either B or C win (where either way the betrayers prefer that to A winning)

15.2305%

10.1852%

19.4445%

Q: Loser drop-out paradox: If B drops out, that switches the winner from A to C. Also (which happens in exactly the same set of elections) "Favorite betrayal"; voters with favorite B, by betraying B, make C win (whom they prefer as the "lesser evil" over current winner A)

12.1583%

9.2014%

19.4445%

And below is the same table, but restricted to elections in which the IRV process
matters, i.e. in which the IRV and plain-plurality winners differ.
(Warning: The error bars are approximately twice as wide as in the tables above.)
This almost always makes pathologies substantially more likely:

X: Would be strategic mistake for more voters of some single type to come

66.3350%

59.1548%

32.2220%

T: Plurality and IRV winners differ

100.0000%

100.0000%

100.0000%

S: Condorcet cycle

18.6193%

25.3526%

0.0000%

Q∪R∪U∪V∪W∪X∪Y∪Z ("total paradox probability")

74.1026%

72.6136%

54.4439%

Both kinds of participation failure simultaneously W∩X

4.8383%

9.0224%

0.0000%

Both kinds of nonmonotonicity simultaneously U∩V

7.6563%

5.9859%

0.0000%

Q∪V: Betraying B makes either B or C win (where either way the betrayers prefer that to A winning)

35.8569%

41.7844%

22.2219%

Q: Loser drop-out paradox: If B drops out, that switches the winner from A to C. Also (which happens in exactly the same set of elections) "Favorite betrayal"; voters with favorite B, by betraying B, make C win (whom they prefer as the "lesser evil" over current winner A)

23.3002%

33.8034%

22.2219%

When IRV-pathologies get more likely when IRV & plurality winners differ
(and they usually do get more likely!), that suggests that IRV, counter-intuitively,
actually is worse than plain-plurality voting – at least
as far as that particular paradox
is concerned, and in 3-way elections.

4. Bugs(?)

Anthony Quas pointed me in the right direction to find a bug in my Quas-generator.
My Dirichlet generator was also buggy due
to some imbecile on Wikipedia publishing an incorrect sorting network.
Both now repaired.

I now agree with the scientific literature values for every number I'm aware of
in these tables which has already been computed in that literature (over a dozen)?
To make that clearer, the below table lists some exact values claimed in the scientific literature
versus our Monte Carlo estimates from the blue table above.
(I may sometimes be listing somebody as "discoverer," who actually was not the first.
Quite a few of these values were independently discovered by several authors.)

5. Example of use

These based on old run at lower precision??? Redo with new hi-precn numbers to get slight changes

Prof. Jack Nagel asked what the "total paradox probabilities" would be if
R were not included as a "paradox." (Note that S and T already were not included.)
His question gives us an excellent opportunity to demonstrate the use of the tables.
The only possible binary codes representing "R plus some subset of S and T, but no other
letters" are

0100000000, 0101000000, 0110000000, and 0111000000.

Of these, the only one which actually occurs with nonzero probability (see the pink table)
is
0101000000
with probabilities
0.0164%, 0.0096%, 0.0000%.
We may therefore obtain Nagel's "no-R total paradox probabilities" by subtracting these
figures from the "total paradox probabilities" we gave in the blue table.
The results are

(24.5873–0.0164≈24.57)%, (13.9847–0.0096≈13.98)%, and 27.50%

in our three different probability models
respectively (error bars≤±0.00025% as usual).
Restricted to elections in which the IRV process matters,
the answers instead are

where note the normalization constants 0.24466 and 0.12326 arise from the
total-T line of the blue table.
The reason these adjustments are so small is that R is a rare paradox, and the occasions where
R arises unaccompanied by any other paradoxes in
{Q, U, V, W, X, Y, Z} – which are the only ones we need to subtract –
are very rare.

6. What about C→∞ candidates instead of C=3?

THEOREM:
As the number C of candidates is made large, the "total paradox probability"
for IRV approaches 100% in all three probability models here.

PROOF (INCOMPLETE IN THE QUAS-1D CASE):
In the Dirichlet model,
Quas 2004 in his theorem 1
showed that the probability of paradox 'U' ("more is less" nonmonotonicity)
approached 100% when C→∞.
Smith 2009 was able to show the same result
in the Random Election Model by
redoing Quas's proof with
appropriate changes.
In the Quas-1D model, a Condorcet winner
generically exists (and always exists aside from the possibility of a 2-way tie)
as a consequence of
Duncan Black's singlepeakedness theorem
and it is the candidate located closest to the midpoint. However, when C→∞ Quas
found that IRV elects the candidate located at x with probability proportional to a certain
probability density ρ(x) with support on (1/6, 5/6) and pictured below.

Therefore IRV falls victim to paradox 'Y' with probability→100%.
Since the "total paradox probability" is at least as great as the probabilities of
these particular paradoxes, it must approach 100%
in all three models, and the proof is complete.
Q.E.D.

WHY THAT PROOF WAS INCOMPLETE IN THE QUAS-1D CASE:
Actually, the above proof is inadequate because Quas's paper did not prove the existence of
ρ(x), although Quas did provide convincing evidence, e.g. the picture above computed by
Monte Carlo, for its existence.
It seems "obvious" that IRV can exhibit no special favortism for the candidate
nearest the midpoint
(who automatically is the Condorcet winner),
as opposed to any of the other candidates located between 0.499 and 0.501
(whose number approaches ∞); indeed the picture indicates, if anything, IRV
is biased against the centermost within this interval.
Therefore it is "obvious" that the probability IRV elects this
one centermost candidate approaches 0. But this "obviousness" is not a "proof."

It would suffice to prove that after IRV has winnowed the C candidates down to, say, C/logC,
the probability→1 (when C→∞) that the centermost candidate has been eliminated.

CONJECTURE:
Paradox 'X' also should occur with probability→100% in the Quas-1D model,
indeed in all three models.

Also see our seperate page
about "spoiler candidates" in IRV and plain plurality voting,
where, among other things, it is argued that in both
voting systems in all three of our probability models,
the chance goes to 100% that your election contains at least one "spoiler"
and hence the probability for "paradox Q" should go to 100% in all three emodels
when C→∞.

7. Sources

The tables here were computed by Warren D. Smith, August 2010, using computer.
Also see the same results
but computed using different random number generator as a safety measure.
[If you average the results from this and that page, you could shrink the error bars
by a factor of √2.]
Smith incorporated results
from the following literature sources into his
computer program:

Beware: Incorrect (but published) results were computed by Crispin Allard and by
Eivind Stensholt (and probably others too; unfortunately I myself had
computed some incorrect values in
CRV web pages). It is quite easy to make your computer get
approximately the right value for
any given paradox probability. Unfortunately, it also is quite easy to mess up
the definitions hence not get it
exactly right. (Also, only good random number generators
passing all 'Supercrush U01' and Marsaglia 'Diehard' tests should be employed;
system-supplied generators usually fail numerous tests in those batteries even as of 2010.)
It is usually rather difficult to make your computer get an
enormously-wrong estimate. The only case I am aware of where that happened was Allard's
paper, which underestimated the nonmonotonicity probability
by 3 orders of magnitude by making 4 seperate errors,
all biased in the same direction, and apparently never doing any simple "sanity checks."

Update (Nov. 2015):
I recently was made aware of the 2009 Dartmouth senior thesis
(pdf) by Joe Ornstein
titled "Instant Runoff Voting's Startling Rate of Failure."
He examines 10000 instant runoff voting elections each in 20 different probabilistic models,
including "spatial models." Ornstein is especially interested in "close elections"
defined as "any election in which three candidates receive more than 25% of
the first-place votes."
Depending which of his 20 probabilistic models he uses,
Ornstein finds that 7% to 33% of his elections are close.
As a fraction of all elections (both close and non-close) 0.6% to 13.5%
feature "type I nonmonotonicity" (our paradox "U"),
depending which of Ornstein's 20 probability models we use.
And Ornstein finds that 13% to 45% of the close elections feature paradox U.
Furthermore, the higher paradox percentages coincide with the probability models
Ornstein considers more realistic.

Miller's conclusions seem consistent with mine and with Ornstein's,
at least on shallow examination.

8. Conclusion

That's incredible. Before I had the sense that IRV was bad because paradoxes could happen
and weren't so rare. Now it's looking like IRV is benign only in those unproblematic elections
where all election methods perform well.
In cases where you actually need a better way of voting, IRV is a crapshoot.
–
William Poundstone, author of
the 2008 popular science book
Gaming the Vote,
after seeing this page (this was his comment by email).